manuscripta mathematica

, Volume 158, Issue 1–2, pp 85–101 | Cite as

A Lie theoretical construction of a Landau–Ginzburg model without projective mirrors

  • E. Ballico
  • S. Barmeier
  • E. GasparimEmail author
  • L. Grama
  • L. A. B. San Martin


We describe the Fukaya–Seidel category of a Landau–Ginzburg model \(\mathrm {LG}(2)\) for the semisimple adjoint orbit of \(\mathfrak {sl}(2, {\mathbb {C}})\). We prove that this category is equivalent to a full triangulated subcategory of the category of coherent sheaves on the second Hirzebruch surface. We show that no projective variety can be mirror to \(\mathrm {LG}(2)\), and that this remains so after compactification.

Mathematics Subject Classification

53D37 14J33 22E60 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



We are grateful to Patrick Clarke for pointing out a significant improvement to an earlier version of this work. We thank Denis Auroux, Lutz Hille, Ludmil Katzarkov, and Sukhendu Mehrotra for helpful suggestions and comments. S. Barmeier is supported by the Studienstiftung des deutschen Volkes. Part of this work was completed during a visit of L. Grama to Chile. We are thankful to the Vice Rectoría de Investigación and Desarrollo Tecnológico of the Universidad Católica del Norte whose support made this visit possible. L. Grama is partially supported by FAPESP Grant 2016/22755-1. E. Gasparim was partially supported by a Simons Associateship ICTP, and Network Grant NT8, Office of External Activities, ICTP, Italy. E. Ballico was partially supported by MIUR and GNSAGA of INdAM (Italy).


  1. 1.
    Atiyah, M.F.: On analytic surfaces with double points. Proc. R. Soc. Lond. Ser. A Math. Phys 247, 237–244 (1958)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Auroux, D.: A beginner’s introduction to Fukaya categories. In: Bourgeois, F., Colin, V., Stipsicz, A. (eds.) Contact and Symplectic Topology, pp. 85–136. Springer, Heidelberg (2016)Google Scholar
  3. 3.
    Auroux, D., Katzarkov, L., Orlov, D.: Mirror symmetry for weighted projective planes and their noncommutative deformations. Ann. Math. 167, 867–943 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ballico, E., Gasparim, E., Grama, L., San Martin, L.A.B.: Some Landau–Ginzburg models viewed as rational maps. Indag. Math. 28, 615–628 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Barmeier, S.: Ph.D. thesis (in preparation)Google Scholar
  6. 6.
    Barmeier, S., Gasparim, E.: Classical deformations of noncompact surfaces and their moduli of instantons. arXiv:1604.01133
  7. 7.
    Fukaya, K., Oh, Y., Ohta, H., Ono, K.: Lagrangian Intersection Floer Theory: Anomaly and Obstruction, Part I. American Mathematical Society/International Press, Somerville (2009)zbMATHGoogle Scholar
  8. 8.
    Gasparim, E., Grama, L., San Martin, L.A.B.: Symplectic Lefschetz fibrations on adjoint orbits. Forum Math. 28, 967–979 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gasparim, E., Grama, L., San Martin, L.A.B.: Adjoint orbits of semi-simple Lie groups and Lagrangian submanifolds. Proc. Edinb. Math. Soc. 60, 361–385 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gorodentscev, A., Kuleshov, S., Rudakov, A.: \(t\)-stabilities and \(t\)-structures on triangulated categories (Russian). Izv. Ross. Akad. Nauk Ser. Mat. 68, 117–150 (2004), translation in Izv. Math. 68, 749–781 (2004)Google Scholar
  11. 11.
    Hille, L., Perling, M.: Tilting bundles on rational surfaces and quasi-hereditary algebras. Ann. Inst. Fourier 64, 625–644 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kashiwara, M., Schapira, P.: Sheaves on Manifolds. Springer, Berlin (1994)zbMATHGoogle Scholar
  13. 13.
    Khovanov, M., Seidel, P.: Quivers, Floer cohomology, and braid group actions. J. Am. Math. Soc. 15, 203–271 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kuznetsov, A., Lunts, V.A.: Categorical resolutions of irrational singularities. Int. Math. Res. Not. 13, 4536–4625 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Orlov, D.: Geometric realizations of quiver algebras. Proc. Steklov Inst. Math. 290, 70–83 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Seidel, P.: More about vanishing cycles and mutation. In: Symplectic Geometry and Mirror Symmetry. Proceedings of the 4th KIAS Annual International Conference, Seoul, South Korea, 14–18 August 2000, pp. 429–465. World Scientific (2001)Google Scholar
  17. 17.
    Tyurina, G.N.: Resolution of singularities of plane deformations of double rational points. Funkc. Anal. i Prilož. 4, 77–83 (1970), translation in Funct. Anal. Appl. 4, 68–73 (1970)Google Scholar
  18. 18.
    Wei, Z.: The full exceptional collections of categorical resolutions of curves. J. Pure Appl. Algebra 220, 3332–3344 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TrentoPovo, TrentoItaly
  2. 2.Mathematisches InstitutWestfälische Wilhelms-Universität MünsterMünsterGermany
  3. 3.Departamento de MatemáticasUniversidad Católica del NorteAntofagastaChile
  4. 4.Imecc – Unicamp, Departamento de Matemática, Rua Sérgio Buarque de Holanda, 651Cidade Universitária Zeferino VazCampinasBrasil

Personalised recommendations